--- a/src/HOL/Library/List_Prefix.thy Thu Dec 06 17:05:44 2007 +0100
+++ b/src/HOL/Library/List_Prefix.thy Thu Dec 06 19:58:21 2007 +0100
@@ -63,6 +63,8 @@
assume "xs \<le> ys @ [y]"
then obtain zs where zs: "ys @ [y] = xs @ zs" ..
show "xs = ys @ [y] \<or> xs \<le> ys"
+ by (metis append_Nil2 butlast_append butlast_snoc prefixI zs)
+(*
proof (cases zs rule: rev_cases)
assume "zs = []"
with zs have "xs = ys @ [y]" by simp
@@ -73,9 +75,12 @@
then have "xs \<le> ys" ..
then show ?thesis ..
qed
+*)
next
assume "xs = ys @ [y] \<or> xs \<le> ys"
then show "xs \<le> ys @ [y]"
+ by (metis order_eq_iff strict_prefixE strict_prefixI' xt1(7))
+(*
proof
assume "xs = ys @ [y]"
then show ?thesis by simp
@@ -85,6 +90,7 @@
then have "ys @ [y] = xs @ (zs @ [y])" by simp
then show ?thesis ..
qed
+*)
qed
lemma Cons_prefix_Cons [simp]: "(x # xs \<le> y # ys) = (x = y \<and> xs \<le> ys)"
@@ -94,19 +100,23 @@
by (induct xs) simp_all
lemma same_prefix_nil [iff]: "(xs @ ys \<le> xs) = (ys = [])"
+by (metis append_Nil2 append_self_conv order_eq_iff prefixI)
+(*
proof -
have "(xs @ ys \<le> xs @ []) = (ys \<le> [])" by (rule same_prefix_prefix)
then show ?thesis by simp
qed
-
+*)
lemma prefix_prefix [simp]: "xs \<le> ys ==> xs \<le> ys @ zs"
+by (metis order_le_less_trans prefixI strict_prefixE strict_prefixI)
+(*
proof -
assume "xs \<le> ys"
then obtain us where "ys = xs @ us" ..
then have "ys @ zs = xs @ (us @ zs)" by simp
then show ?thesis ..
qed
-
+*)
lemma append_prefixD: "xs @ ys \<le> zs \<Longrightarrow> xs \<le> zs"
by (auto simp add: prefix_def)
@@ -114,28 +124,34 @@
by (cases xs) (auto simp add: prefix_def)
theorem prefix_append:
- "(xs \<le> ys @ zs) = (xs \<le> ys \<or> (\<exists>us. xs = ys @ us \<and> us \<le> zs))"
+ "(xs \<le> ys @ zs) = (xs \<le> ys \<or> (\<exists>us. xs = ys @ us \<and> us \<le> zs))"
apply (induct zs rule: rev_induct)
apply force
apply (simp del: append_assoc add: append_assoc [symmetric])
+ apply (metis append_eq_appendI)
+(*
apply simp
apply blast
+*)
done
lemma append_one_prefix:
- "xs \<le> ys ==> length xs < length ys ==> xs @ [ys ! length xs] \<le> ys"
- apply (unfold prefix_def)
+ "xs \<le> ys ==> length xs < length ys ==> xs @ [ys ! length xs] \<le> ys"
+by (unfold prefix_def)
+ (metis Cons_eq_appendI append_eq_appendI append_eq_conv_conj eq_Nil_appendI nth_drop')
+(*
apply (auto simp add: nth_append)
apply (case_tac zs)
apply auto
done
-
+*)
theorem prefix_length_le: "xs \<le> ys ==> length xs \<le> length ys"
by (auto simp add: prefix_def)
lemma prefix_same_cases:
- "(xs\<^isub>1::'a list) \<le> ys \<Longrightarrow> xs\<^isub>2 \<le> ys \<Longrightarrow> xs\<^isub>1 \<le> xs\<^isub>2 \<or> xs\<^isub>2 \<le> xs\<^isub>1"
- apply (simp add: prefix_def)
+ "(xs\<^isub>1::'a list) \<le> ys \<Longrightarrow> xs\<^isub>2 \<le> ys \<Longrightarrow> xs\<^isub>1 \<le> xs\<^isub>2 \<or> xs\<^isub>2 \<le> xs\<^isub>1"
+by (unfold prefix_def) (metis append_eq_append_conv2)
+(*
apply (erule exE)+
apply (simp add: append_eq_append_conv_if split: if_splits)
apply (rule disjI2)
@@ -150,43 +166,45 @@
apply (insert append_take_drop_id [of "length xs\<^isub>1" xs\<^isub>2])
apply simp
done
+*)
+lemma set_mono_prefix: "xs \<le> ys \<Longrightarrow> set xs \<subseteq> set ys"
+by (auto simp add: prefix_def)
-lemma set_mono_prefix:
- "xs \<le> ys \<Longrightarrow> set xs \<subseteq> set ys"
- by (auto simp add: prefix_def)
-
-lemma take_is_prefix:
- "take n xs \<le> xs"
- apply (simp add: prefix_def)
+lemma take_is_prefix: "take n xs \<le> xs"
+by (unfold prefix_def) (metis append_take_drop_id)
+(*
apply (rule_tac x="drop n xs" in exI)
apply simp
done
-
+*)
lemma map_prefixI:
"xs \<le> ys \<Longrightarrow> map f xs \<le> map f ys"
- by (clarsimp simp: prefix_def)
+by (clarsimp simp: prefix_def)
lemma prefix_length_less:
"xs < ys \<Longrightarrow> length xs < length ys"
- apply (clarsimp simp: strict_prefix_def)
+by (clarsimp simp: strict_prefix_def prefix_def)
+(*
apply (frule prefix_length_le)
apply (rule ccontr, simp)
apply (clarsimp simp: prefix_def)
done
-
+*)
lemma strict_prefix_simps [simp]:
"xs < [] = False"
"[] < (x # xs) = True"
"(x # xs) < (y # ys) = (x = y \<and> xs < ys)"
- by (simp_all add: strict_prefix_def cong: conj_cong)
+by (simp_all add: strict_prefix_def cong: conj_cong)
-lemma take_strict_prefix:
- "xs < ys \<Longrightarrow> take n xs < ys"
- apply (induct n arbitrary: xs ys)
- apply (case_tac ys, simp_all)[1]
+lemma take_strict_prefix: "xs < ys \<Longrightarrow> take n xs < ys"
+apply (induct n arbitrary: xs ys)
+ apply (case_tac ys, simp_all)[1]
+apply (metis order_less_trans strict_prefixI take_is_prefix)
+(*
apply (case_tac xs, simp)
apply (case_tac ys, simp_all)
- done
+*)
+done
lemma not_prefix_cases:
assumes pfx: "\<not> ps \<le> ls"
@@ -195,17 +213,17 @@
| (c2) a as x xs where "ps = a#as" and "ls = x#xs" and "x = a" and "\<not> as \<le> xs"
| (c3) a as x xs where "ps = a#as" and "ls = x#xs" and "x \<noteq> a"
proof (cases ps)
- case Nil
- then show ?thesis using pfx by simp
+ case Nil thus ?thesis using pfx by simp
next
case (Cons a as)
- then have c: "ps = a#as" .
-
+ hence c: "ps = a#as" .
show ?thesis
proof (cases ls)
- case Nil
+ case Nil thus ?thesis by (metis append_Nil2 pfx c1 same_prefix_nil)
+(*
have "ps \<noteq> []" by (simp add: Nil Cons)
from this and Nil show ?thesis by (rule c1)
+*)
next
case (Cons x xs)
show ?thesis
@@ -234,12 +252,14 @@
then have npfx: "\<not> ps \<le> (y # ys)" by simp
then obtain x xs where pv: "ps = x # xs"
by (rule not_prefix_cases) auto
-
+ show ?case by (metis Cons.hyps Cons_prefix_Cons npfx pv r1 r2)
+(*
from Cons
have ih: "\<And>ps. \<not>ps \<le> ys \<Longrightarrow> P ps ys" by simp
show ?case using npfx
by (simp only: pv) (erule not_prefix_cases, auto intro: r1 r2 ih)
+*)
qed
@@ -250,16 +270,16 @@
"(xs \<parallel> ys) = (\<not> xs \<le> ys \<and> \<not> ys \<le> xs)"
lemma parallelI [intro]: "\<not> xs \<le> ys ==> \<not> ys \<le> xs ==> xs \<parallel> ys"
- unfolding parallel_def by blast
+unfolding parallel_def by blast
lemma parallelE [elim]:
- assumes "xs \<parallel> ys"
- obtains "\<not> xs \<le> ys \<and> \<not> ys \<le> xs"
- using assms unfolding parallel_def by blast
+assumes "xs \<parallel> ys"
+obtains "\<not> xs \<le> ys \<and> \<not> ys \<le> xs"
+using assms unfolding parallel_def by blast
theorem prefix_cases:
- obtains "xs \<le> ys" | "ys < xs" | "xs \<parallel> ys"
- unfolding parallel_def strict_prefix_def by blast
+obtains "xs \<le> ys" | "ys < xs" | "xs \<parallel> ys"
+unfolding parallel_def strict_prefix_def by blast
theorem parallel_decomp:
"xs \<parallel> ys ==> \<exists>as b bs c cs. b \<noteq> c \<and> xs = as @ b # bs \<and> ys = as @ c # cs"
@@ -275,17 +295,25 @@
then obtain ys' where ys: "ys = xs @ ys'" ..
show ?thesis
proof (cases ys')
- assume "ys' = []" with ys have "xs = ys" by simp
+ assume "ys' = []"
+ thus ?thesis by (metis append_Nil2 parallelE prefixI snoc.prems ys)
+(*
+ with ys have "xs = ys" by simp
with snoc have "[x] \<parallel> []" by auto
then have False by blast
then show ?thesis ..
+*)
next
fix c cs assume ys': "ys' = c # cs"
+ thus ?thesis
+ by (metis Cons_eq_appendI eq_Nil_appendI parallelE prefixI same_prefix_prefix snoc.prems ys)
+(*
with snoc ys have "xs @ [x] \<parallel> xs @ c # cs" by (simp only:)
then have "x \<noteq> c" by auto
moreover have "xs @ [x] = xs @ x # []" by simp
moreover from ys ys' have "ys = xs @ c # cs" by (simp only:)
ultimately show ?thesis by blast
+*)
qed
next
assume "ys < xs" then have "ys \<le> xs @ [x]" by (simp add: strict_prefix_def)
@@ -301,18 +329,16 @@
qed
qed
-lemma parallel_append:
- "a \<parallel> b \<Longrightarrow> a @ c \<parallel> b @ d"
- by (rule parallelI)
- (erule parallelE, erule conjE,
- induct rule: not_prefix_induct, simp+)+
+lemma parallel_append: "a \<parallel> b \<Longrightarrow> a @ c \<parallel> b @ d"
+by (rule parallelI)
+ (erule parallelE, erule conjE,
+ induct rule: not_prefix_induct, simp+)+
-lemma parallel_appendI:
- "\<lbrakk> xs \<parallel> ys; x = xs @ xs' ; y = ys @ ys' \<rbrakk> \<Longrightarrow> x \<parallel> y"
- by simp (rule parallel_append)
+lemma parallel_appendI: "\<lbrakk> xs \<parallel> ys; x = xs @ xs' ; y = ys @ ys' \<rbrakk> \<Longrightarrow> x \<parallel> y"
+by simp (rule parallel_append)
lemma parallel_commute: "(a \<parallel> b) = (b \<parallel> a)"
- unfolding parallel_def by auto
+unfolding parallel_def by auto
subsection {* Postfix order on lists *}
@@ -322,12 +348,12 @@
"(xs >>= ys) = (\<exists>zs. xs = zs @ ys)"
lemma postfixI [intro?]: "xs = zs @ ys ==> xs >>= ys"
- unfolding postfix_def by blast
+unfolding postfix_def by blast
lemma postfixE [elim?]:
- assumes "xs >>= ys"
- obtains zs where "xs = zs @ ys"
- using assms unfolding postfix_def by blast
+assumes "xs >>= ys"
+obtains zs where "xs = zs @ ys"
+using assms unfolding postfix_def by blast
lemma postfix_refl [iff]: "xs >>= xs"
by (auto simp add: postfix_def)
@@ -380,42 +406,37 @@
then show "xs >>= ys" ..
qed
-lemma distinct_postfix:
- assumes "distinct xs"
- and "xs >>= ys"
- shows "distinct ys"
- using assms by (clarsimp elim!: postfixE)
+lemma distinct_postfix: "distinct xs \<Longrightarrow> xs >>= ys \<Longrightarrow> distinct ys"
+by (clarsimp elim!: postfixE)
-lemma postfix_map:
- assumes "xs >>= ys"
- shows "map f xs >>= map f ys"
- using assms by (auto elim!: postfixE intro: postfixI)
+lemma postfix_map: "xs >>= ys \<Longrightarrow> map f xs >>= map f ys"
+by (auto elim!: postfixE intro: postfixI)
lemma postfix_drop: "as >>= drop n as"
- unfolding postfix_def
- by (rule exI [where x = "take n as"]) simp
+unfolding postfix_def
+by (rule exI [where x = "take n as"]) simp
-lemma postfix_take:
- "xs >>= ys \<Longrightarrow> xs = take (length xs - length ys) xs @ ys"
- by (clarsimp elim!: postfixE)
+lemma postfix_take: "xs >>= ys \<Longrightarrow> xs = take (length xs - length ys) xs @ ys"
+by (clarsimp elim!: postfixE)
lemma parallelD1: "x \<parallel> y \<Longrightarrow> \<not> x \<le> y"
- by blast
+by blast
lemma parallelD2: "x \<parallel> y \<Longrightarrow> \<not> y \<le> x"
- by blast
+by blast
lemma parallel_Nil1 [simp]: "\<not> x \<parallel> []"
- unfolding parallel_def by simp
+unfolding parallel_def by simp
lemma parallel_Nil2 [simp]: "\<not> [] \<parallel> x"
- unfolding parallel_def by simp
+unfolding parallel_def by simp
-lemma Cons_parallelI1:
- "a \<noteq> b \<Longrightarrow> a # as \<parallel> b # bs" by auto
+lemma Cons_parallelI1: "a \<noteq> b \<Longrightarrow> a # as \<parallel> b # bs"
+by auto
-lemma Cons_parallelI2:
- "\<lbrakk> a = b; as \<parallel> bs \<rbrakk> \<Longrightarrow> a # as \<parallel> b # bs"
+lemma Cons_parallelI2: "\<lbrakk> a = b; as \<parallel> bs \<rbrakk> \<Longrightarrow> a # as \<parallel> b # bs"
+by (metis Cons_prefix_Cons parallelE parallelI)
+(*
apply simp
apply (rule parallelI)
apply simp
@@ -423,7 +444,7 @@
apply simp
apply (erule parallelD2)
done
-
+*)
lemma not_equal_is_parallel:
assumes neq: "xs \<noteq> ys"
and len: "length xs = length ys"
@@ -435,7 +456,6 @@
next
case (2 a as b bs)
have ih: "as \<noteq> bs \<Longrightarrow> as \<parallel> bs" by fact
-
show ?case
proof (cases "a = b")
case True